65.17.6 problem 13 (a)

Internal problem ID [15849]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 7. Systems of First-Order Differential Equations. Exercises page 329
Problem number : 13 (a)
Date solved : Thursday, October 02, 2025 at 10:28:38 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )-2 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{2} \left (x \right )-y_{1} \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=1 \\ y_{2} \left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.164 (sec). Leaf size: 118

ode:=[diff(y__1(x),x) = 3*y__1(x)-2*y__2(x), diff(y__2(x),x) = -y__1(x)+y__2(x)]; 
ic:=[y__1(0) = 1, y__2(0) = -1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (x \right ) &= \left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x}+\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x} \\ y_{2} \left (x \right ) &= -\frac {\left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x} \sqrt {3}}{2}+\frac {\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x} \sqrt {3}}{2}+\frac {\left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x}}{2}+\frac {\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 79

ode={D[ y1[x],x]==3*y1[x]-2*y2[x],D[ y2[x],x]==-y1[x]+y2[x]}; 
ic={y1[0]==1,y2[0]==-1}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)&\to \frac {1}{2} e^{-\left (\left (\sqrt {3}-2\right ) x\right )} \left (\left (1+\sqrt {3}\right ) e^{2 \sqrt {3} x}+1-\sqrt {3}\right )\\ \text {y2}(x)&\to -\frac {1}{2} e^{-\left (\left (\sqrt {3}-2\right ) x\right )} \left (e^{2 \sqrt {3} x}+1\right ) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 66

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-3*y__1(x) + 2*y__2(x) + Derivative(y__1(x), x),0),Eq(y__1(x) - y__2(x) + Derivative(y__2(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
 

\[ \left [ y^{1}{\left (x \right )} = - C_{1} \left (1 - \sqrt {3}\right ) e^{x \left (2 - \sqrt {3}\right )} - C_{2} \left (1 + \sqrt {3}\right ) e^{x \left (\sqrt {3} + 2\right )}, \ y^{2}{\left (x \right )} = C_{1} e^{x \left (2 - \sqrt {3}\right )} + C_{2} e^{x \left (\sqrt {3} + 2\right )}\right ] \]